Summary of Computer doctoral thesis: Some extensions of the complex fuzzy inference system for decision support problem

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Research objectives of the thesis: Research the theories of complex fuzzy sets, complex fuzzy logic and measures based on complex fuzzy sets; research and development of fuzzy inference system based on complex fuzzy sets; research applied techniques to reduce rules, optimize fuzzy rules in complex fuzzy inference system; research on how to represent rules based on fuzzy knowledge graphs to reduce inference computation time for the test set and deal with the cases where the new dataset is not present in the training data set.

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MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ------------------------------- LUONG THI HONG LAN SOME EXTENSIONS OF THE COMPLEX FUZZY INFERENCE SYSTEM FOR DECISION SUPPORT PROBLEM Major: Computer science Code: 9 48 01 01 SUMMARY OF COMPUTER DOCTORAL THESIS Ha Noi - 2021
The doctoral thesis was completed at Graduate University of Science and Technology – Vietnam Academy of Science and Technology Supervisor 1: Assoc. Prof. Dr. Le Hoang Son Supervisor 2: Assoc. Prof. Dr. Nguyen Long Giang Reviewer 1: Reviewer 2: Reviewer 3: This doctoral thesis will be defended at the Board of Examiners of Graduate University of Science and Technology, Vienam Academy of Science and Technology on hour….., date….. month….. 2021 This doctoral thesis can be explored at: - Library of the Graduate University of Science and Technology - National Library of Vietnam
1 PREFACE Fuzzy set (FS) [1] proposed by Zadeh in 1965 is considered as an effective tool to solve the problems with uncertain properties. Various extensions and operations of FS have been presented in recent years [2-6]. One of the most important techniques in FS is Fuzzy Inference System (FIS), which is widely applied in many decision-making and classification/prediction problems such as green supplier selection, personnel selection, company strategy, etc. In these applications, FIS was used to generate a set of fuzzy rules to detect, predict or classify objects such as lung cancer detection, detection of diabetes mellitus, heart disease prediction, evaluation of green supply chain management performance, penetration index estimation in rock mass [7-13]. An extended version of FIS embedded with neural network and gradient-based learning is the Adaptive Neuro Fuzzy Inference System (ANFIS) [14], which also demonstrated good performance in coronary artery disease prognosis, estimating thermal conductivity enhancement of metal and metal oxide, flood prediction, etc. [15- 21]. Recently, with the boost up of various decision-making problems inspired by time variants or phase changes, an extension of the Fuzzy Set namely the Complex Fuzzy Set (CFS) with new membership functions including both the amplitude and the phase terms has been proposed [37]. CFS has been applied with much concentration by new fuzzy aggregation operators, complex fuzzy soft information, distance measures, and complex fuzzy concept lattice [37]–[43]. The advantage of CFS is the capability to model phenomena and events by the phase term to show their overall progress within a given context. For an example, in order to validate whether blood pressure of a patient is ‘High’ or ‘Low’, a sample of 30 measured times is recorded, and the mean and variance are computed. Using the fuzzification of FIS in the CFS, the blood pressure of the patient can be easily measured within the recorded period as ‘Low’ amplitude with ‘Low’ phase (i.e. small mean and variance values). If the blood pressure is measured in a specific time stamp, a wrong decision may be given out. Another example of the problem of disease diagnosis: if only based on the disease attribute values without considering other attributes, the diagnosis result will be inaccurate, because the disease conclusion depends not only on each disease attribute value, but also need to consider factors related to that disease. Moreover, there are also many scenarios that involves a phase term, which is encountered in data with a periodic trend, such as rainfall recorded in a region, or the sound waves produced by a musical instrument. It is therefore evident, that complex numbers must be given a place in the literature of fuzzy inference system as well. This is therefore the main motive of this thesis. The ordinary fuzzy inference systems such as the Mamdani, Sugeno and Tsukamoto systems and various versions of the ANFIS architectures are only able to handle phenomena that are not periodic or seasonal. In order to handle time-series data in time-periodic phenomena, FISs and ANFISs employ two general strategies: 1) ignore the information related to the phase term; 2) represent the amplitude and phase terms separately using two fuzzy sets. This would cause loss of information and produce unreliable results (if information related to the phase terms are ignored), distortion of information, and a reduction in computational efficiency (if information related to the amplitude and phase are represented separately) as it becomes more time-consuming due to the increased number of sets that need to be dealt with. Complex fuzzy inference systems are considered to be an effective tool for solving problems with periodicity and uncertainty. The first system introduced by Ramot [44] is called Complex Fuzzy Logic System, which is developed from the usual fuzzy logic system but replaces the fuzzy set and the implication rule by its version in complex plane. Another study by Man et al. [45] is based on the combination of inductive learning
2 with inference systems in complex fuzzy sets. Another version of embedded learning with neural fuzzy network on CFS set called Adaptive Neural Complex Fuzzy Inference System (ANCFIS) was introduced by Chen et al. [46]. Then two improvements of ANCFIS with the aim of increasing the computational speed are also given in [47-48]. In the other words, most of the so-called CFISs that have been proposed in the literature are not truly complex systems. From the existing studies on complex fuzzy systems, complex fuzzy systems still have some limitations as follows: - Complex fuzzy systems have not provided an overall procedure for building complex fuzzy inference systems for decision support systems. -The rule set are only established based on experience, based on logical thinking, without mentioning the problem of optimizing complex fuzzy inference systems. - Complex fuzzy systems have not been studied to apply to new datasets that are not included in the training data when generating the inference model. - The complex fuzzy t-normal and t-normal operators have not yet been pay attention and applied in decision support systems. Research objectives of the thesis. The thesis focuses on researching and applying complex fuzzy inference system to the problem of decision support system, as following: 1) Research the theories of complex fuzzy sets, complex fuzzy logic and measures based on complex fuzzy sets 2) Research and development of fuzzy inference system based on complex fuzzy sets 3) Research applied techniques to reduce rules, optimize fuzzy rules in complex fuzzy inference system 4) Research on how to represent rules based on fuzzy knowledge graphs to reduce inference computation time for the test set and deal with the cases where the new dataset is not present in the training data set. The layout of the thesis consists of four main chapters, Introduction, Conclusion and list of references. The Introduction presents an overview of the research problem, the reasons for choosing the topic, objectives and research content of the thesis. The Conclusion summarizes the achieved results of the thesis and researchs in future. The main content chapters are organized as follows Chapter 1 presents the basic concepts and background knowledge that will be used in the next chapters, such as fuzzy set, complex fuzzy set, fuzzy measure, complex fuzzy measure and related studies on inference system based on complex fuzzy set in recent years. On that basis, the thesis analyzes the remaining problems, clearly the research motivations of the thesis: using complex fuzzy inference system for solving problems to support decision making. In addition, the experimental data sets in the thesis with the measures used for experimental evaluation are also detailed in this first chapter. Chapter 2 presents two main research results: the first is the definition of complex fuzzy t-norm and t- norm operations and the example to use these operations in decision support system; the second is to develop Mamdani inference system on complex fuzzy set. The choice of aggregation operator, the methods of determining the rule firing strength and defuzzification methods are presented clearly in this chapter. Chapter 3 proposes M-CFIS-R model that is a new Mamdani Complex Fuzzy Inference System with Rule Reduction Using Complex Fuzzy Measures in Granular Computing. Several fuzzy similarity measures such
3 as Complex Fuzzy Cosine Similarity Measure (CFCSM), Complex Fuzzy Dice Similarity Measure (CFDSM), and Complex Fuzzy Jaccard Similarity Measure (CFJSM) together with their weighted versions are proposed. Those measures are integrated into the M-CFIS-R system by the idea of Granular Computing such that only important and dominant rules are being kept in the system The problem of reducing the rule system in the Mamdani complex fuzzy inference system is the content considered in chapter 3. Based on the theory of granular computing, the thesis proposes three complex fuzzy measures and combined these measures with granular computing to optimize the rule system in the complex fuzzy inference system Mamdani proposed in chapter 2 (M-CFIS-R complex fuzzy inference system). Numerical examples and experimental results have also demonstrated the effectiveness of the problem of rule in the Mamdani complex fuzzy inference system. If chapter 3 only focuses on the problem of rule reduction and optimization in the training phase, then chapter 4 focus on improving the testing set by applying the theory of fuzzy knowledge graphs. In addition, the thesis also proposes some exxtesions of M-CFIS-R: Sugeno Complex Fuzzy Inference Systems (S-CFIS-R), Tsukamoto Complex Fuzzy Inference Systems (T-CFIS-R), Complex fuzzy measures and Complex fuzzy integrals in M-CFIS-R. Finally, the conclusion section presents the contributions of the thesis, development direction and issues of concern of the author. CHAPTER 1. INTRODUCTION 1.1. Introduction Fuzzy set theory and Complex Fuzzy Set are approaches for representing and processing vagueness found abundantly in the real world. 1.2. Problem of Fuzzy Inference Systems in Decision Support Systems General process of the method of using fuzzy systems in decision support systems Figure 1.1. Fuzzy Inference Systems in Decision Support Systems Firstly, based on the training sample data, a rule generation procedure is applied to generate a set of fuzzy rules. This rule set is the center of the collection of rules and knowledge extracted from the training data set. Next, for each new input is applied to each rule and compute the outputs. A process that aggregates results from rules to produce final value. Finally, at the decision-making step, this value is adjusted and normalized to make the final decision. 1.3. An overview of related works 1.3.1. Fuzzy Inference System Fuzzy Inference System (FIS) is an a popular computational framework based on the concept of fuzzy theory and commonly applied when construct decision support model. There are three types of FIS: Mamdani FIS, Sugeno FIS(or Takagi – Sugeno), Tsukamoto FIS.
4 1.3.2. Complex Fuzzy Inference Systems 1.3.2.1 Complex Fuzzy Logic System of Ramot Ramot proposed a complex fuzzy logic system (CFLS) that consists of three stages: The fuzzification module; The fuzzy inference stage and The defuzzification process. In CFLS, Ramot et al. did not outline any specific method of defuzzification to reduce the complex fuzzy outputs into crisp outputs. 1.3.2.2. CANFIS Model of Li and Jang Li và Jang proposed a FIS based on CFSs called the Complex Neuro-Fuzzy Inference System (CANFIS). This system is however not truly complex in nature, as the real and imaginary parts of the input membership functions are dealt with separately using two type-1 fuzzy sets. Separating the real and imaginary parts also leads to an increased number of rules which makes this system computationally expensive. 1.3.2.3. ANCFIS model of Chen et al ANCFIS structure proposed by Chen et al in 2010, similar to the complex valued neural network structure. The ANCFIS models by Man [12] and Chen [13] used vector dot-product for the aggregation stage and treated the complex-valued inputs as real values, thereby enabling them to obtain scalar values for the dot product. This would not be possible if the inputs are indeed treated as complex values, as the dot product of two complex numbers is a complex number and not a scalar value. The ANCFIS system is therefore not truly complex as the outputs of the system will not be representative of the periodicity of the elements. 1.3.2.4. OtherFuzzy Inference Systems on Complex Fuzzy Set Besides the existing studies, complex fuzzy sets have also been interested and developed by many research groups. Malekzadeh and Akbarzadeh [27] proposed another inference system based on complex fuzzy sets, called the Complex-valued Adaptive Neuro-fuzzy Inference System (CANFIS), which is a hybrid of CFIS and fuzzy neural networks. However, the authors did not present any method to deal with the defuzzification of the complex-valued outputs to crisp values, and chose to only consider the real part of the output. Deshmukh et al. introduced a complex fuzzy logic module and applied this to the design process of a fuzzy microprocessor using the VLSI approach. However, the authors did not implement rule interference and did not provide a valid defuzzification module. 1.3.3. Remaining problems of the current CFIS system From previous studies, most of the so-called CFISs that have been proposed in the literature are not truly complex systems. In order to handle time-series data in time-periodic phenomena, FISs and ANFISs employ two general strategies: 1) ignore the information related to the phase term; 2) represent the amplitude and phase terms separately using two fuzzy sets. This would cause loss of information and produce unreliable results (if information related to the phase terms are ignored), distortion of information, and a reduction in computational efficiency (if information related to the amplitude and phase are represented separately) as it becomes more time-consuming due to the increased number of sets that need to be dealt with. 1.4. Theoretical basic 1.4.1. A fuzzy set The concept of fuzzy sets was introduced by Lotfi A.Zadel in 1965 [1] with the aim of describing the concepts of "unclear sets" in the study of uncertain factors. 1.4.2. Complex Fuzzy Set Complex Fuzzy Set is characterized by a membership function  A  x  thatlies within the unit circle in the complex plane and has the form:
5  A  x   rA  x .e j  x , j  1 A (0.1) Where the amplitude rA  x  and phase  A  x  are both real-valued with condition rA  x    0,1 and  A  x   (0,2 ] . 1.4.3. Some basic operations of CFS: 1.4.3.1 Complement of a Complex Fuzzy Set Let A and B be two complex fuzzy sets, and let:  A ( x)  rA ( x)e jA ( x ) and B ( x)  rB ( x)e jB ( x ) , the complement of A (denoted as A ) and is specified by the function:  A  ( x,  A ( x)) | x U   ( x, rA ( x)e j A ( x ) ) | x U  (1.4) Where rA ( x)  1  rA ( x) and A ( x)  2  A ( x) . 1.4.3.2. Union and intersection of two Complex Fuzzy Sets  The union of A and B is denoted as A  B : A  B  ( x,  A B ( x)) | x U   ( x, rA B ( x)e jAB ( x ) ) | x U  (1.5)  ( x,  rA ( x)  rB ( x) e jAB ( x ) ) | x U  Where  is t-conorm, for instance, rA B ( x)  max rA ( x), rB ( x)  Intersection of two Complex Fuzzy Sets, A and B (denoted as A  B ): A  B  ( x,  A B ( x)) | x U   ( x, rA B ( x)e jAB ( x ) ) | x U  (1.6)  ( x,  rA ( x)  rB ( x) e jAB ( x ) ) | x U  Where rA B ( x)  min rA ( x), rB ( x) and  A B ( x)  min  A ( x), B ( x)  Where  is t-norm, for example, Min-operator. 1.4.4. Complex Fuzzy Logic A complex fuzzy logic system (CFLS) consists of a complex fuzzy rule (CFL) base on complex fuzzy set to form a complex fuzzy logic system. A CFL represents a complex fuzzy implication relation between complex fuzzy propositions p and q, where p ∼ “X is A” and q ∼ “Y is B,” respectively. A complex fuzzy implication is then defined as  A B  x, y    A  x    B  y  (1.14) 1.4.5. Fuzzy measures and complex fuzzy measures Definition: [44] A distance of complex fuzzy sets is  :  F * U   F * U     0,1 for any A, B and C  F * U  if satisfies: o   A, B   0,   A, B   0 khi và chỉ khi A  B o   A, B     B, A (1.16) o   A, B     A, C     C , B  where F U  is the set of all complex fuzzy sets in U * 1.5. Experiment datasets 1.5.1. Benchmark datasets To illustrate the proposed models, the thesis uses five commonly available Benchmark datasets taken from the UCI Machine Learning Depository, including Wisconsin Breast Cancer Diagnosis (WBCD), Diebetes, Wine Quality, CardiotocoGraphy and Arrhythmia Datasets.
6 1.5.2. Real dataset The real dataset received from the Gangthep Hospital and Thai Nguyen National Hospital, Vietnam. including 4156 patients that come to the hospital for examination of their liver function. Based on these results, the physician may ask the patient to perform additional examinations aiming to improve the diagnosis. 1.5.3. Experimental evaluation measures The metrics used to evaluate the effectiveness of proposed model for the decision support system include: Accuracy, Precision measure, Recall measure and computational time. 1.6. Chapter conclusion Chapter 1 presents some basic concepts of complex fuzzy set theory and existing fuzzy inference systems and complex fuzzy inference systems, overview of research on fuzzy inference systems based on complex fuzzy sets. These contents will be the background knowledge and use in the next chapters of the thesis. Chương 2. CONSTRUCT MAMDANI COMPLEX FUZZY INFERENCE SYSTEM (M-CFIS) 2.1. Overview This chapter introduces Mamdani complex fuzzy inference system with details of the components as well as implementation and operators. Moreover, the complex fuzzy t-norm and t-conorm also are proposed and applied in decision support problems. 2.2. Propose complex fuzzy t-norm and t-conorm 2.2.1. t-norm and t-conorm This section presents the brief of t-norm and t-conorm operators. 2.2.2. Complex fuzzy t-norms and t-conorms Definition 2.3. Let J : 0,1  0,1  0,1 be a mapping where  0,1 is the unit complex disk in the set of complex numbers. Then J is called a complex T-norm if the following conditions hold for all p, q, r   0,1 where p, q, r are the complex fuzzy membership grade with p  p1e j1 , q  q1e j2 , r  r1e j3 (1) J  p, q   J  q, p  , (2) J  p, q   J  p, r  , nếu q  r , (3) J  p, J  q, r    J  J  q, p  , r  , (4) J  p,1  p Definition 2.4. Let J * :  0,1   0,1   0,1 be a mapping where  0,1 is the unit complex disk in the set of complex numbers. Then J * is called a complex T -conorm if the following conditions hold for all p, q, r   0,1 where p  p1e j1 , q  q1e j2 , r  r1e j3 are the complex fuzzy membership grade: (1) J *  p, q   J *  q , p  , (2) J *  p, q   J *  q, r  , nếu q  r , (3) J *  p, J *  q, r    J *  J *  q, p  , r  , (4) J *  p,0   p J  p, p   p for Definition 2.5. If the complex fuzzy t-norm function J  p, q  is continuous and all p   0,1 then it is called an Archimedean complex fuzzy t-norm. If an Archimedean complex fuzzy t- norm is strictly increasing with respect to each variable for p, q   0,1 then it is called a strict Archimedean complex fuzzy t-norm. Definition 2.6. If a complex fuzzy t-conorm function J *  p, q  is continuous and J *  p, p   p for all p   0,1 then it is called an Archimedean complex fuzzy t-conorm. If an Archimedean complex fuzzy t-
7 conorm is strictly increasing with respect to each variable p, q   0,1 then it is called a strict Archimedean complex fuzzy t-conorm. Definition 2.7. Let  : 0,1  0,1  0,1 ,  is called a negation function if: (1) N  0   1, N 1  0 (2) N  p   N  q  when p  q Definition 2.8. A negation function  is called strict if: (1)  is continuous and (2) strictly decreasing: N  p   N  q  if p  q for all p, q   0,1 Definition 2.9. A negation function  is called involutive if it is strict and N  N  p    p for all p  0,1 . 2.2.3. An example in multi-criteria decision making algorithm In this section, we apply the t-norm operators to develop a multi-criteria decision making (MCDM) algorithm, which consists of the following steps: Step 1. Consider a MCDM problem where there are m alternatives   i  1,2,..., m  and n criteria  k  k  1, 2,..., n  . The decision maker constructs the decision matrix    yik mn where yik represents the degree that the decision maker prefers the alternative  i with respect to the criterion  k . The weights of the  j    criteria are expressed as the CFNs  k   k e k ,  k  1, 2,..., n  , where  k indicates the amplitude function/degree that the decision maker prefers criterion  k and  k indicates the phase term/degree. Step 2. Transform the decision matrix    yik mn into the normalized decision matrix D   ik mn , yik where ik  , i  1,..., m; k  1,..., n max yik Step 3.i ,kUtilize the operators in Example 2.3 to compute the Lukasiewicz complex fuzzy T-norms. Step 4: Summ up the complex membership grades. Step 5: Consider the highest score as the candidate for the best ranking.
8 2.3. Mamdani complex fuzzy inference system (M-CFIS) 2.3.1. Proposes Mamdani complex fuzzy inference system Figure 2.1. The framework of M-CFIS 2.3.2. Some choices use in M-CFIS 2.3.2.1. Complex fuzzy membership function In M-CFIS model, the classic complex membership fuction is used, as follows:   x   r  x .ei  x whith r  x    0,1 and   x    0, 2  representing the amplitude and phase terms of the membership grade. 2.3.2.2. Operations used in the M- CFIS In this research, the operations that will be used in our M-CFIS are given below: 1. The minimum T-norm is used for calculating the firing strength of a complex fuzzy rule with AND connecting the antecedents. 2. The maximum T-conorm is used for calculating the firing strength of a complex fuzzy rule with OR connecting the antecedents. 3. The Mamdani implication rule for complex fuzzy sets is used to calculate the values of the consequent of each complex fuzzy rule:    x  B  y   i 2  A   A B  x, y    rA  x  .rB  y   .e .  2 2  2.3.2.3. Vector aggregation for CFSs M-CFIS proposed the dot product between complex-valued vectors which is as given:
9 wp . Ap  x   wp . Ap  x   rp' e  i 'p r Ap  xe  i Ap  x    rp' rAp  x  e  i  'p  Ap  x      rp' rAp  x  cos  'p   Ap  x   i sin  'p   Ap  x     2.3.2.4. Aggregation of the output distribution The output distribution: D  y   1  y    2  y   ...   k  y  with  p  y  are complex functions. This way, we can be sure of obtaining a truly complex CFIS in which the information pertaining to the phase are not disregarded but taken into consideration in every step of the decision-making process. 2.3.3. Structure of the Mamdani CFIS The proposed Mamdani CFIS consists of six stages which must be completed before an output is obtained. Each of these individual stages are as given below: Stage 1: Determine a set of complex fuzzy rules Establish a set of complex fuzzy rules of the form: CFR1: If xm1,1 is A1,1 O1,1 xm1,2 is A1,2 O1,2 … O1, n1 1 xm1,n1  is 𝐴1,𝑛1 then y is C1 CFR2: If xm 2,1 is A2,1 O2,1 xm 2,2 is A2,2 O2,2 … O2, n2 1 xm 2,n2  is A2,n2 then y is C 2 … … … … CFRk: If xm k ,1 is Ak ,1 Ok ,1 xm k ,2 is Ak ,2 Ok ,2 … Ok , nk 1 xm k ,nk  is Ak , nk then y is C k in which: (a) m  p, q  1, 2,..., n with 1  m  p,1  m  p, 2   ...  m  p, n p   n     (b)  Ap ,q xm p ,q   rAp ,q xm p ,q  e   i Ap ,q xm p ,q  , with rAp ,q :  0,1 and  Ap ,q :   0,2  . iC p  y  (c) C p  y   rC p  y  e , with rC p :  0,1 and C p :   0, 2  . (d) T0 is a T-norm, and S 0 is the S-norm (i.e. the T-conorm) that corresponds to 𝑇0 . (e) f p :  0, 2  p   0, 2  , with f p  2 ,2 ,...,2   2 n i p (f)  p   p e , where  p  f p A  p ,1  x    ,  x    ,  x    ,..., m p ,1 Ap ,2 m p ,2 Ap ,3 m p ,3 Ap ,n p  x    . m p,np (i) Op , q  and iff N p , q  T0 (ii) O p , q  or iff N p ,q  S0 Stage 2: Fuzzification. This stage involves finding the fuzzified input membership function values:   A p ,q a     r a   e m p ,q Ap , q m p ,q i Ap ,q am p ,q  with p, q Stage 3: Establishing the rule firing strength. i p Compute the firing strengths for each complex fuzzy rule wp   p e , where:  p  N p ,n p 1 ...N  N r p ,2 p ,1 Ap ,1  x    , r  x    , r  x   ..., r m p ,1 Ap ,2 m p ,2 Ap ,3 m p ,3 Ap ,n p  x    m p ,np   p  f p A p ,1  x    , m p ,1 Ap ,2  x    ,  x    ,...,  x    m p ,2 Ap ,3 m p ,3 Ap ,n p m p,np Stage 4: Calculating the consequence of the complex fuzzy rules   We form the consequent of CFR p :  p  y   U 0  p , rC p  y  e   ig0  p ,C p  y  Stage 5: Aggregation for the output distribution The output distribution is defined as: D  y   1  y    2  y   ...   k  y 
10 Stage 6: Defuzzification. Choose a function  : F  ,  , Determine the value of the output: yop    D  . For instance, we   y D  y  dy may choose the approximation of   D    using the trapezoidal rule, for all f  F  , .  D  y  dy  2.4. Experimental results T his section aims to evaluate performance of proposed M-CFIS with Mamdani fuzzy inference system (M-FIS) in benchmark UCI datasets UCI and real medical dataset from Gangthep Hospital and Thai Nguyen National Hospital. The experimental results are described in Figure 2.2, Figure 2.3 and Figure 2.4. Figure 2.2. Performance on the WBCD Figure 2.3. Performance on the Diebetes Figure 2.4. Performance on the Liver Chương 3. MAMDANI COMPLEX FUZZY INFERENCE SYSTEM WITH RULE REDUCTION (M-CFIS-R) 3.1. Overview One of the limitations of the existing M-CFIS is the rule base that may be redundant to a specific dataset and based on the caculation of the strength rule. Thus, the rule set in M-CFIS may be still redundant. In order to handle the problem, this chapter presents the improvement of optimizing the rule system of M- CFIS by combine some similarity measures and granular computing.
11 3.2. Propose complex fuzzy measure 3.2.1. Complex Fuzzy Cosine Similarity Measure Definition 3.1. Assume that there are two complex fuzzy sets S1  rS1  x  e và S 2  rS2  x  e jS1 ( x ) jS2 ( x ) in S for all x  X , both amplitude and phase term in [0,1]. A Complex Fuzzy Cosine Similarity Measure (CFCSM) between S1 and S 2 : 1 n a1a2  b1b2 CCFS   (3.1) n k 1  a1    b1    a2    b2  2 2 2 2  where a1  Re rS1  x  e jS1 ( x ) ;  b1  Im rS1  x  e jS1 ( x ) ; a 2   Re rS2  x  e jS2 ( x ) ; b 2   Im rS2  x  e jS2 ( x )  Definition 3.2. Weighted Complex Fuzzy Cosine Similarity Measure (WCNCSM) Assume that there are two complex fuzzy sets S1  rS1  x  e và S 2  rS  x  e jS1 ( x ) jS2 ( x ) 2 in S for all x  X . A Weighted Complex Fuzzy Cosine Similarity Measure between S1 and S 2 : n a1a2  b1b2 n CWCFS   k với  k 1 (3.2)  a1    b1    a2    b2  2 2 2 2 k 1 k 1 3.2.2. Complex Fuzzy Dice Similarity Measure Definition 3.3. Assume that there are two complex fuzzy sets S1  rS1  x  e và S 2  rS2  x  e jS1 ( x ) jS2 ( x ) in S for all x  X , both amplitude and phase term in [0,1]. A Complex Fuzzy Dice Similarity Measure (CFCSM) between S1 and S 2 : 1 n 2 a1b1a2b2 DCFS   n k 1 a1b1  a2b2 (3.3)  where a1  Re rS1  x  e jS1 ( x ) ;  b1  Im rS1  x  e jS1 ( x ) ; a 2   Re rS2  x  e jS2 ( x ) ; b 2   Im rS2  x  e jS2 ( x )  Definition 3.4. Weighted Complex Fuzzy Dice Similarity Measure (WCFDSM) Assume that there are two complex fuzzy sets S1  rS1  x  e và S 2  rS2  x  e jS1 ( x ) jS2 ( x ) in S for all x  X , both amplitude and phase term in [0,1]. A Weighted Complex Fuzzy Dice Similarity Measure between S1 and S 2 : n  2 a1b1a2b2  n (3.4) DWCFS   k   với  k 1 k 1  a1b1  a2b2  k 1 3.2.3. Complex Fuzzy Jaccard Similarity Measure Definition 3.5. Assume that there are two complex fuzzy sets S1  rS  x  e và S 2  rS  x  e jS1 ( x ) jS2 ( x ) 1 2 in S for all x  X , both amplitude and phase term in [0,1]. A Complex Fuzzy Jaccard Similarity Measure (CFCSM) between S1 and S 2 : 1 n a1b1a2b2 J CFS   n k 1  a1b1  a2b2   a1b1 . a2b2   (3.5)  Với a1  Re rS1  x  e jS1 ( x ) ;  b1  Im rS1  x  e jS1 ( x ) ; a 2   Re rS2  x  e jS2 ( x ) ; b 2   Im rS2  x  e jS2 ( x )  Definition 3.6. Weighted Complex Fuzzy Jaccard Similarity Measure (WCFJSM) Assume that there are two complex fuzzy sets S1  rS  x  e và S 2  rS  x  e jS1 ( x ) jS2 ( x ) 1 2 in S for all x  X , both amplitude and phase term in [0,1]. A Weighted Complex Fuzzy Jaccard Similarity Measure between S1 and S 2 :
12   n 3.6  n a1b1a2b2 JWCFS   k   với 1    a1b1  a2b2   a1b1 . a2b2   k k 1 k 1   3.3. Proposed M-CFIS-R System 3.3.1. Main ideas M-CFIS-R is devided into two main parts: The Training phase in order to create the original complex fuzzy rule base and improve it by the Granular Computing with Complex Fuzzy Measure; The Testing phase used to test the performance of the rule system in Training phase. 3.3.2. Training Figure 3.1. Training diagram for the proposed model 3.3.2.1. Real and Imaginary Data Selection. From the Training data, we build the real and imaginary data as follows: The real data are defined as the original data values. The imaginary data at P of attribute Q is determined as: var.P(row)+ var.Q(column) where var.P (row) is the variance in row at row P and var.Q (column) is the variance according to the column in column Q .
13 3.3.2.2. Fuzzy C-Means (FCM) The Fuzzy C-Means clustering method is used for dividing the data according to each attribute into several groups. The number of clusters specified for each attribute is different based on the semantic value of the attribute. 3.3.2.3 Granular Complex Fuzzy Measures Assume that the outputs of three similarity measures are three corresponding squared matrices whose elements are the correlations between pairs of complex fuzzy rules ( D1 , D 2 , D 3 ). Then, we determine the final degree of similarity between complex fuzzy rules based on the aggregation: Fij  a1Dij1  a2 Dij2  a3 Dij3 For each set of labels, we obtain Fij (l ) to be determined a1 (l ), a2 (l ),..., ae (l ) : Dt / l Dt / l D / t t at (l )   ij i 1 j i 1 Dt / l For rules other than labels, then Fij  0 . From these, we obtain the matrix F . A new complex fuzzy rule base is found from F by removing rules having a high or maximal degree of similarity within a group. Then, we proceed to the next steps to evaluate the performance of the new rule system. In cases that the performance of the new complex fuzzy rule base is worse than that of the current rule, we return to the steps of computing the complex fuzzy measures and granular computing for the new complex fuzzy rule base. The iteration stops either when the performance of the new complex fuzzy rule base is better than that of the current base or the cardinality of rules according to any label is equal to 1. 3.3.3. Testing We perform a similar procedure with M-CFIS [23] for testing the performance of the system with the reduced complex fuzzy rule base found in the Training phase. 3.4. Experiments 3.4.1. Experimental Results on the Benchmark UCI Datasets Using 3-fold cross-validation method, the values of criteria obtained by applying M-CFIS and M- CFIS-R on the UCI datasets are visually presented in Figures 3.3 and 3.4, respectively. (a) (b) (c) (d) (e) Figure 3.3. Performance on the WBCD dataset Figure 3.3 shows the results of applying M-CFIS and M-CFIS-R on the first dataset-WBCD. The accuracy, the recall and Precision of M-CFIS-R in the training data and testing data is higher than that of M-
14 CFIS. The computation time of these methods can be considered as equal with 36 rules less than the result of M-CFIS. (b) (c) (a) 200 Số lượng luật 100 106 101 0 M-CFIS M-CFIS-R (d) (e) Figure 3.4. Performance on the Diebetes dataset The values of validity indices (Figure 5a,b) obtained from M-CFIS-R are higher than those of M-CFIS by more than 1% and with small SD. But, the running time (Figure 5c) of M-CFIS-R is higher than that of M- CFIS by only 0.02 s on the training data and 0.086 s on the testing data. The average number of rules in Figure 5d of M-CFIS-R is 5 rules less than that of M-CFIS, with SD of 0.94. 3.4.2. Experimental Results on the Real Datasets On the real datasets, the classification quality evaluation between our proposed method M-CFIS-R and M-CFIS is shown in Figures 3.5. (a) (b) (c) 900 850 Số lượng luật 800 839 750 770 700 M-CFIS M-CFIS-R (d) (e) Figure 3.5. Performance on the Liver dataset Figure 3.5 shows the performance of M-CFIS-R and M-CFIS on the Liver dataset. It is clear that the accuracy, the recall and precision values of M-CFIS-R on the training data is higher than that of M-CFIS.
15 Although the recall of M-CFIS-R on the testing data is 0.4% smaller than that of M-CFIS, the SD is very small (only 0.03). This is caused by the decreasing in number of rules. On the Liver dataset, the number of rules in M-CFIS-R is 69 less than that of M-CFIS. This is the reason for M-CFIS-R being more time-consuming than M-CFIS. Chương 4. EXTENSION MAMDANI COMPLEX FUZZY INFERENCE SYSTEM WITH KNOWLEDGE GRAPH (M-CFIS-FKG) 4.1. Overview The M-CFIS-R model has been designed to utilize granular computing with complex similarity measures to reduce the rule base so as to gain better performance in decision-making problems. However M- CFIS-R has some limitations as follows: (1), testing data are checked by matching with each rule in the rule base, which leads to a high cost of computational time; (2) if the testing data contain records that are not inferred by the rule base, the output cannot be generated ; (3) The M-CFIS-R model works based on the Mamdani inference model, which needs to be developed on the Sugeno and Tsukamoto inference models. (4) Other complex fuzzy integral or measure concepts also need to be considered. Therefore, it is for these reasons that in Chapter 4, the thesis presents a new approach based on the knowledge graph to overcome the limitations of the M-CFIS-R model that has proposed in Chapter 3. 4.2. Some extensions of M-CFIS-R 4.2.1. Sugeno and Tsukamoto complex fuzzy inference system  Sugeno complex fuzzy inference system (S-CFIS-R): The steps in S-CFIS-R are as follows Step 1. Rule formation. A complex fuzzy rule CFRi can be expressed as follows: CFRi : If xm,1 is Ai ,1 Oi ,1 xm ,2 is Ai ,2 Oi ,2 … Oi , ni 1 xm,ni is Ai , ni then f i ; where: Ai ,1 ,..., Ai , ni are the complex fuzzy sets taken by the rule antecedent variables; Oi ,1 ,..., Oi ,ni 1 the complex T-norm or T-conorm operator depending on practical applications and f i is a polynomial function of the rule’s consequent.. Step 2: Fuzzification. Find the fuzzified input values with the complex fuzzy membership function hóa. Step 3: Aggregation for firing strengths. Calculate the firing strength value w m for each rule by the function w m   m e j  m ;  Step 4: Determine the consequence value of the complex fuzzy rules: zm  f m am,1 , am,2 ,..., am,nm  Step 5: Aggregation for the output distribution. Denote zm  vm ei m and wm  zm  ( mvm ) ei g0 ( m , m ) for all m. The output is obtained by using the weighted average formula as given below: k w  z  w2  z2  ...  wk  zk w m  zm yop  1 1  m 1 w1  w2  ...  wk k w m 1 m  Tsukamoto complex fuzzy inference system (T-CFIS-R): The inference process on Tsukamoto complex fuzzy inference system is similar to Sugeno complex fuzzy inference system. Each consequence of rules in Tsukamoto complex fuzzy inference system has specified by a monotonous function on complex fuzzy set. Thus, the inference outputs of each rule are obtained based on the predicate. Finally, the final is calculated by a weighted average formula (same to S-CFIS-R).
16 4.2.2. Complex fuzzy measures based on set theory Definition 4.1. Given a non-empty complex fuzzy set C on universe discourse X . A subset CF of CF (C ) is an algebra of complex fuzzy sets on C if it satisfies the following conditions: (1) 1 , C  CF , (2) If P  CF , then C \ P  CF (3) If P, Q  CF , then P  Q  CF Definition 4.2. Given the complex fuzzy measurable space  C , CF  . A mapping r : CF  H is defined as complex fuzzy measure on  C , CF  if it satisfies: (1) r 1  .e j 1   0 and r  C .e jC   1 (2) rD  x .e jD  x  rE  x .e jE  x for any x  X , with D, E  X and D  E Definition 4.3. Given  C , CF  ,  D, CG  are complex fuzzy measurable spaces and a mapping k : CF  CG . Mapping k is called an isomorphism between  C , CF  and  D, CG  if the following conditions holds: (1) k is a bijective mapping with k 1   1 (2) k  P  Q   k  P   k  Q  and k  C \ P   D \ k  P  with P, Q  CF (3) Existing a bijective mapping l : Dom  C   Dom  D  with P  x   k  P   l  x   , P  CF and x  Dom  C  Definition 4.4. Given complex fuzzy measures spaces (C, CF , r ,  ) and ( D, CG, r * , * ) . A mapping k : CF  CG is called an isomorphism mapping between (C, CF , r ,  ) and ( D, CG, r * , * ) if the following conditions are satisfies (1) k is an isomorphism mapping between  C , CF  and  D, CF  . (2) r  P  .e j  P   r *  k  P   .e j  P  with P  CF * Definition 4.5. A complex fuzzy space (C, CF , r ,  ) is cardinal space if the following holds: (1) P  CF , then l   P   CF ,   for any P  CF and any permutation l on Dom C .  (2) r  P  .e j  P   r l   P  .e  j l   P    4.2.3. Complex fuzzy integral Definition 4.6. Given a complex fuzzy measure space  C , CF , r ,   with X  Dom(C ) , a mapping l : X  H and CF - measure P . Complex fuzzy integral - of k on P is calculated as follows: P fdrP  x  .e j P  x     QCFP xsupp Q  l  x  rQ  q  .e jQ  q   với CFP   CFP \ 1  4.3.3.1. Complex fuzzy integral Definition 4.7. Given a complex fuzzy measurable space  C , CF  and X  Supp  C  , an algebra of sets SP on X is a crisp representation of the algebra CF if and only if there is Q  CF for any P  SP such that: supp  Q   P and if supp  Q *  P for Q*  CF , then Q*  Q . 4.3.3.2. A relation to the Sugeno integral Theorem 4.4. Given a complete divisible residual lattic U , a complex fuzzy measure space  C , CF , r ,   with X  Dom  C  and mapping l : X  H .  j b  *   ldrP  b  .e jP  b     b  r *P 1  b  .e  Kb   P1 Then we have: bH   Kb  P  where  r * ,  *  is the inner complex fuzzy measure on  C , CF  C   . 4.3.3.3. Properties of the complex fuzzy integral Theorem 4.6. Given  C , CF , r ,   with X  Dom  C  .
17 If D  CF  X  with C  D, CG  A lg  D  and C is crisp, then  : CG  H that is specified by the below function: P  p  .e j P  p    PdrP  p .e jP  p  is the complex fuzzy measure on  D, CG  . 4.3. Propose Mamdani complex fuzzy inference systems with fuzzy knowledge graph M-CFIS-FKG 4.3.1. Main ideas With the aim of making the inference process in Testing faster we extend the M-CFIS-R as follows. The initial data are divided into 3 parts namely Training, Validation, and Testing. From the Training data, we build the real and imaginary data. Using the M-CFIS-R model, we obtain the complex fuzzy rule base with suitable and effective number of rules. Then, we construct Fuzzy Knowledge Graph (FKG) from the rule base and represent it by an adjacency matrix. In the Testing, we design the Fast Inference Search Algorithm (FISA) to derive the outputs from the FKG. 4.3.2. Construct fuzzy knowledge graph Suppose that we have the following complex fuzzy rule base with X1, X2, …Xm are attributes of dataset. We gradually build FKG for each rule Rt , with t  1, k . For each pair  X i , X j  , 1  i  j  m in rule   Rt , let us construct an edge as Value  X i   Value X j where Value  X i  is the linguistic variable in attribute X i . With each  X i , Label  ,1  i  m , an edge Value  X i   Labelt is constructed where Labelt is the label of rule t. Suppose that Aijt is the weight of the edge Value  X i   Value  X j  in rule t với t  1, k , 1  i  j  m X i relation with X j in rule t , then Aijt : Aijt  (1) R The weight Bilt is the relationship of the attribute X i to the label l where t  1, k , 1  i  j  m , l  1, C . Then Bilt is caculated as   X relation with label l in rule t Bilt    Aijt  * i (2)   R
18 Figure 4.1. The Training of the proposed model Figure 4.2. The Testing